RStudio Exercise 1: Tools and methods

The name of the course is Introduction to Open Data Science and we are focusing to language R, RStudio, GitHub and Markdown. You can find my Github repository here.


RStudio Exercise 2: Analysis

Introduction to the data

After the data wrangling exercise the new data set is found from the data folder. The set is based on data that was collected from course Introduction to Social Statistics, fall 2014 - in Finnish. The survey was conducted 3.12.2014 - 10.1.2015 by Kimmo Vehkalahti.

student2014 <- read.table("data/learning2014.txt", header = TRUE)
dim(student2014)
## [1] 166   7

The student data includes 7 variables and 166 rows.

str(student2014)
## 'data.frame':    166 obs. of  7 variables:
##  $ gender  : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
##  $ age     : int  53 55 49 53 49 38 50 37 37 42 ...
##  $ attitude: int  37 31 25 35 37 38 35 29 38 21 ...
##  $ deep    : num  3.58 2.92 3.5 3.5 3.67 ...
##  $ stra    : num  3.38 2.75 3.62 3.12 3.62 ...
##  $ surf    : num  2.58 3.17 2.25 2.25 2.83 ...
##  $ points  : int  25 12 24 10 22 21 21 31 24 26 ...

Variables deep, strat and surf are combination variables from the original survey data.

summary(student2014)
##  gender       age           attitude          deep            stra      
##  F:110   Min.   :17.00   Min.   :14.00   Min.   :1.583   Min.   :1.250  
##  M: 56   1st Qu.:21.00   1st Qu.:26.00   1st Qu.:3.333   1st Qu.:2.625  
##          Median :22.00   Median :32.00   Median :3.667   Median :3.188  
##          Mean   :25.51   Mean   :31.43   Mean   :3.680   Mean   :3.121  
##          3rd Qu.:27.00   3rd Qu.:37.00   3rd Qu.:4.083   3rd Qu.:3.625  
##          Max.   :55.00   Max.   :50.00   Max.   :4.917   Max.   :5.000  
##       surf           points     
##  Min.   :1.583   Min.   : 7.00  
##  1st Qu.:2.417   1st Qu.:19.00  
##  Median :2.833   Median :23.00  
##  Mean   :2.787   Mean   :22.72  
##  3rd Qu.:3.167   3rd Qu.:27.75  
##  Max.   :4.333   Max.   :33.00

From 166 survey respondents 56 was men and 110 was females. The mean of age was 25,5 years. Oldes respondet was 55 and youngest 17 years old.

Graphical output

Variables differ between genders. Distributions are different in age, attitude and surf (surfface). Three highest correlation between variables are:

  • points-attitude
  • surf-deep
  • surf-attitude

Explanatory variables

Three variables

## 
## Call:
## lm(formula = points ~ attitude + stra + surf, data = student2014)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -17.1550  -3.4346   0.5156   3.6401  10.8952 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 11.01711    3.68375   2.991  0.00322 ** 
## attitude     0.33952    0.05741   5.913 1.93e-08 ***
## stra         0.85313    0.54159   1.575  0.11716    
## surf        -0.58607    0.80138  -0.731  0.46563    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.296 on 162 degrees of freedom
## Multiple R-squared:  0.2074, Adjusted R-squared:  0.1927 
## F-statistic: 14.13 on 3 and 162 DF,  p-value: 3.156e-08
  • In this linear regression model points are the target variable and attitude, strategy (stra) and surfface (surf) are explanatory variables.

  • Residuals of the model are between ~ -17.2 and ~10.9 when median is 0.52. I assume that errors are normally distributed but distribution needs confirmation.

  • Attitude is the only variable that has a very good significance in this model. p-value of stra and surf is too high to be even slightly significance.

  • Variables estimated coefficient is ~0.34 and it’s standard error is clearly smaller (~0.057). Other explanatory variables have not significance in this model.

  • Residual standard error is high in relation to a first and third quantiles of residuals.

  • This linear regression model of three explanatory variables explains ~19.3% of the points.

One variable

## 
## Call:
## lm(formula = points ~ attitude, data = student2014)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -16.9763  -3.2119   0.4339   4.1534  10.6645 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 11.63715    1.83035   6.358 1.95e-09 ***
## attitude     0.35255    0.05674   6.214 4.12e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared:  0.1906, Adjusted R-squared:  0.1856 
## F-statistic: 38.61 on 1 and 164 DF,  p-value: 4.119e-09
  • Estimated coefficient of explanatory variable attitude is ~0.35. This means that when attitude rises one point target variable (points) grows 0.35 times.

  • Attitude p-value (0.00000000412) shows that variable is very significant in this linear regression model

  • This model explains 18.6% of the points (target variable)

Graphical model validation

Residuals vs Fitted This plot shows that there is no pattern between residuals. There is a constant variance among errors. One can confirm that the assumption of constant variance of errors is valid.

Normal Q_Q Normal QQ-plot shows that the errors of the model are normally distributed.

Residual vs Leverage Last plot of the graphical model validation shows that the impact of the singel observation is moderate. Model includes some outliers but the leverage of singel observation don’t compromise the validation of the model.


RStudio exercise 3: Logistic regression

Introduction to the data

Using Data Mining To Predict Secondary School Student Alcohol Consumption. Fabio Pagnotta, Hossain Mohammad Amran Department of Computer Science,University of Camerino

https://archive.ics.uci.edu/ml/datasets/STUDENT+ALCOHOL+CONSUMPTION

Data is rolled into one from Math course and Portuguese language course datasets. After the data wrangling exercise the new data set is found from the data folder.

alc <- read.table("data/student_alc.txt", header = TRUE)
dim(alc)
## [1] 382  35

The student data includes 35 variables and 382 rows.

  • The variables not used for joining the two data have been combined by averaging (including the grade variables)
  • ‘alc_use’ is the average of ‘Dalc’ and ‘Walc’
  • ‘high_use’ is TRUE if ‘alc_use’ is higher than 2 and FALSE otherwise

High and low alcohol consumption and other variables

In this analysis I’m going to study the relationship of high/low alcohol consumption between sex and the following variables:

Variable Type Description
age numeric student’s age
studytime numeric, scale 1-4 weekly study time
freentime numeric, scale 1-5 free time after school
absence numeric number of school absences

First these relationships are observed from tables and graphics. Hypothesis are as follows:

Age

The age of high consumption of alcohol may differ between sex. The development of charcter differs between young people and this may affect on habbits of alcohol consumption.

H0: Age don’t affect on alcohol consumption

H1: There is difference in between level of alcohol consumption and age

#a jitterplot of high_use, sex and age
g1 <- ggplot(alc, aes(x = high_use, y = age, col = sex))
g1 + geom_jitter() + ggtitle("Age by alcohol consumption and sex")

It seems that there is randomnes of sex and age in both alcohol consumption groups.

#a boxplot of high_use, sex and age
g1 <- ggplot(alc, aes(x = high_use, y = age, col = sex))
g1 + geom_boxplot() + ggtitle("Age by alcohol consumption and sex") + xlab("High consumption group") + ylab("Age of student")

Means however show that young male students have lower mean of age in low consumption group and females in high consumption group.

Study time

High alcohol consumption may be related to time spend in studies because one can’t do both at the same time at least not successfully.

H0: Alcohol consumption do not affect on weekly study time

H1: Alcohol consumption affects on weekly study time (numeric: 1 - <2 hours, 2 - 2 to 5 hours, 3 - 5 to 10 hours, or 4 - >10 hours)

#barplot about study time
ggplot(alc, aes(studytime, fill = high_use)) + geom_bar(position = "fill") +
  ggtitle("Barplot about study time grouped by high_use")+ xlab("Study time") + ylab("Propotions of students") + scale_y_continuous(name = waiver(), breaks = waiver(), minor_breaks = waiver(), labels = waiver(), limits = NULL, expand = waiver(), na.value = NA_real_, trans = "identity")

It seems that there is less high users in those students groups who spend more time in studying (3-4) than in those who spend less time in studying (1-2).

Free time

High alcohol consumption may be related to free time so that studets who have more free time are consuming more alcohol that studets who haven’t as much free time.

H0: Amount of free time do not affect on alcohol consumption among students

H1: Amount of free time affects on alcohol consumption among students

#barplot about free time
ggplot(alc, aes(freetime, fill = high_use)) + geom_bar(position = "fill") +
  ggtitle("Barplot about free time grouped by high_use")+ xlab("Free time") + ylab("Propotions of students") + scale_y_continuous(name = waiver(), breaks = waiver(), minor_breaks = waiver(), labels = waiver(), limits = NULL, expand = waiver(), na.value = NA_real_, trans = "identity")

It seems that there are more students that are consuming alcohol high amounts in those students groups that have more free time than in those who have not as much free time.

Absences

High consumption of alcohol may cause absences.

H0: High consumption of alcohol do not affect on absences

H1: High consumption of alcohol does affect on absences

#a boxplot of high_use and absences
g1 <- ggplot(alc, aes(x = high_use, y = absences, fill = high_use))
g1 + geom_boxplot() + ggtitle("Absences by alcohol consumption") + xlab("High consumption group") + ylab("Absences")

It seems that the differences between alcohol consumption groups in absences are small. Mean of absences is higher in high consumption group but it may not be significant.

Logistic regression model

#the model with glm()
m <- glm(high_use ~ age + studytime + freetime + absences, data = alc, family = "binomial")
summary(m)
## 
## Call:
## glm(formula = high_use ~ age + studytime + freetime + absences, 
##     family = "binomial", data = alc)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.9649  -0.8267  -0.6238   1.0990   2.2861  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -4.39240    1.79306  -2.450 0.014299 *  
## age          0.18543    0.10313   1.798 0.072183 .  
## studytime   -0.51842    0.15867  -3.267 0.001086 ** 
## freetime     0.33015    0.12430   2.656 0.007907 ** 
## absences     0.07856    0.02269   3.463 0.000535 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 465.68  on 381  degrees of freedom
## Residual deviance: 423.18  on 377  degrees of freedom
## AIC: 433.18
## 
## Number of Fisher Scoring iterations: 4

From the fitted model one can see that all explanatory variables except age are statistically significant with p-value < 0,01. Variable absence is also statistically sisgnificant with p-value < 0,001. It seems that age doesn’t explain whether or not a student is a high user of alcohol.

Odds ratio and confidence intervals

# compute odds ratios (OR)
OR <- coef(m) %>% exp
# compute confidence intervals (CI)
CI <- confint(m) %>% exp
## Waiting for profiling to be done...
# print out the odds ratios with their confidence intervals
cbind(OR, CI)
##                     OR        2.5 %    97.5 %
## (Intercept) 0.01237102 0.0003481843 0.3998333
## age         1.20373817 0.9852330457 1.4775972
## studytime   0.59545916 0.4321414645 0.8061535
## freetime    1.39118359 1.0938389338 1.7826010
## absences    1.08172440 1.0368164005 1.1336048

Odds ratio (OR) and the 95% confidence interval (CI) shows that those students who have a low study time are almost two times as likely to be a high user of alcohol than those studets who have higher study time. Students that have more freetime are also more likely to be a high user of alcohol. Also absences are positively correlated with high use of alcohol. Confidence interval shows that age is not statistically significant (because the interval contains 1) and other variables are.

Predictive power of the model

Predictive power of the final logistic regression model is calculated without the statistically insignificant variable age.

#the model with glm() and without the age variable  
m_final <- glm(high_use ~ studytime + freetime + absences, data = alc, family = "binomial")
summary(m_final)
## 
## Call:
## glm(formula = high_use ~ studytime + freetime + absences, family = "binomial", 
##     data = alc)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.9420  -0.8332  -0.6450   1.1266   2.1537  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -1.34105    0.56337  -2.380 0.017293 *  
## studytime   -0.50496    0.15691  -3.218 0.001290 ** 
## freetime     0.32626    0.12379   2.636 0.008401 ** 
## absences     0.08324    0.02262   3.680 0.000233 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 465.68  on 381  degrees of freedom
## Residual deviance: 426.47  on 378  degrees of freedom
## AIC: 434.47
## 
## Number of Fisher Scoring iterations: 4
#predict and add the answer and the prediction to the data (alc)
probabilities <- predict(m_final, type = "response")
alc <- mutate(alc, probability = probabilities)
alc <- mutate(alc, prediction = probabilities > 0.5)

#tabulate the target variable versus the prediction
table("High use" = alc$high_use, "Prediction" = alc$prediction)
##         Prediction
## High use FALSE TRUE
##    FALSE   254   14
##    TRUE     93   21

Table shows that the model predict 254 true negatives, 21 true positives, 14 false negatives and 93 false postives. This is sometimes called “confusion table”

table("High use" = alc$high_use, "Prediction" = alc$prediction) %>% prop.table() %>% addmargins()
##         Prediction
## High use      FALSE       TRUE        Sum
##    FALSE 0.66492147 0.03664921 0.70157068
##    TRUE  0.24345550 0.05497382 0.29842932
##    Sum   0.90837696 0.09162304 1.00000000

Propabilities of the same table shows that 90,8% is predicted to be false but only 66,5% of them is correct. 9,2% is predicted to be true but 5,5% of them realy are students with high use of alcohol.

#a plot of 'high_use' versus 'probability' in 'alc'
g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))
g + geom_point()

Average number of wrong predictions

#defining a loss function (mean prediction error)
loss_func <- function(class, prob) {
  n_wrong <- abs(class - prob) > 0.5
  mean(n_wrong)
}

#calling loss_func to compute the average number of wrong predictions in the (training) data
loss_func(class = alc$high_use, prob = alc$probability)
## [1] 0.2801047

The average number of wrong predictions in trainig data is 28%.

Cross validation

#computing the average number of wrong predictions in the (training) data
#loss_func(class = alc$high_use, prob = alc$probability)
#K-fold cross-validation
library(boot)
cv <- cv.glm(data = alc, cost = loss_func, glmfit = m_final, K = 10)
#average number of wrong predictions in the cross validation
cv$delta[1]
## [1] 0.2827225

10-fold cross-validation gives good estimate of the actual predictive power of the model. Low value = good.


RStudio exercise 4: Clustering and classification

Introduction to the data

In this exercise we use Boston data from MASS-library. This dataset contains information collected by the U.S Census Service concerning housing in the area of Boston Mass. Data includes 14 variables and 506 rows.

## [1] 506  14
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
variable description
crim per capita crime rate by town
zn proportion of residential land zoned for lots over 25,000 sq.ft.
indus proportion of non-retail business acres per town
chas Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
nox nitrogen oxides concentration (parts per 10 million)
rm average number of rooms per dwelling
age proportion of owner-occupied units built prior to 1940
dis weighted mean of distances to five Boston employment centres
rad index of accessibility to radial highways
tax full-value property-tax rate per $10,000
ptratio pupil-teacher ratio by town
black 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town
lstat lower status of the population (percent)
medv median value of owner-occupied homes in $1000

Graphical overview of the data

Plot matrix of the data

There are some very intresting distributions fo variables in the plot matrix. Variable rad has high and low values so the plot shows that the values are consenrated either side of the plot. VAriable *

Plotted correlation matrix

Plotted correlation matrix shows that there is some high correlation between variables:

  • Correlation is quite clear between industrial areas (indus) and nitrogen oxides (nox). Industry adds pollution in the area. Industry seems to correlate also with variablrs like age, dis, ras and tax.

  • Nitrogen oxides (nox) correlations are very similar with industry (indus)

  • Crime rate (crim) seems to correlate with good accessibilitty to radial highways (rad) and value property (tax).

  • Old houses (age) and employment centers have also something common

summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00

Scaled data

All the variables are numerical so we can use scale()-function to scale whole data set.

##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv        
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865
## [1] "matrix"

Scaling the data makes variables look as if they are in the same range. Variables like black and tax were before scaling hundred fold compared to some other variables.

Creating a new categorical variable crime

Variable crim is the base of the new categorical variable crime.

categories quantile points
low 0%-25%
med_low 25%-50%
med_high 50%-75%
high 75%-100%

Quantile points of the variable crim

##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610
## crime
##      low  med_low med_high     high 
##      127      126      126      127
##        zn               indus              chas              nox         
##  Min.   :-0.48724   Min.   :-1.5563   Min.   :-0.2723   Min.   :-1.4644  
##  1st Qu.:-0.48724   1st Qu.:-0.8668   1st Qu.:-0.2723   1st Qu.:-0.9121  
##  Median :-0.48724   Median :-0.2109   Median :-0.2723   Median :-0.1441  
##  Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.04872   3rd Qu.: 1.0150   3rd Qu.:-0.2723   3rd Qu.: 0.5981  
##  Max.   : 3.80047   Max.   : 2.4202   Max.   : 3.6648   Max.   : 2.7296  
##        rm               age               dis               rad         
##  Min.   :-3.8764   Min.   :-2.3331   Min.   :-1.2658   Min.   :-0.9819  
##  1st Qu.:-0.5681   1st Qu.:-0.8366   1st Qu.:-0.8049   1st Qu.:-0.6373  
##  Median :-0.1084   Median : 0.3171   Median :-0.2790   Median :-0.5225  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4823   3rd Qu.: 0.9059   3rd Qu.: 0.6617   3rd Qu.: 1.6596  
##  Max.   : 3.5515   Max.   : 1.1164   Max.   : 3.9566   Max.   : 1.6596  
##       tax             ptratio            black             lstat        
##  Min.   :-1.3127   Min.   :-2.7047   Min.   :-3.9033   Min.   :-1.5296  
##  1st Qu.:-0.7668   1st Qu.:-0.4876   1st Qu.: 0.2049   1st Qu.:-0.7986  
##  Median :-0.4642   Median : 0.2746   Median : 0.3808   Median :-0.1811  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 1.5294   3rd Qu.: 0.8058   3rd Qu.: 0.4332   3rd Qu.: 0.6024  
##  Max.   : 1.7964   Max.   : 1.6372   Max.   : 0.4406   Max.   : 3.5453  
##       medv              crime    
##  Min.   :-1.9063   low     :127  
##  1st Qu.:-0.5989   med_low :126  
##  Median :-0.1449   med_high:126  
##  Mean   : 0.0000   high    :127  
##  3rd Qu.: 0.2683                 
##  Max.   : 2.9865

Train and test sets

Training set contains 80% of the data. 20% is in the test set.

##   [1] 389 185 205 136 385 249 150 262 384 119  95 478 495 157 280   8 469
##  [18] 191 475 398 139  36 476  47 256  86 179  75 405 367  62 257 193 501
##  [35] 426  13  90  39 146 407 499 438 122 296  17 187 470 274 152 369 473
##  [52] 400 123 177 474 110 453 313 154 463 153  57  73 236 425 423 278 432
##  [69] 305 240 314  87 190  21 344 133 323 115  16 121 445 409 441  67 144
##  [86]  20 415  76  44 408 419 472 489 135 208 422 108 243  32 285 447 358
## [103] 372 365  25 189 293  69  93 209 265 373 272 371 304  68 271 128  46
## [120] 413 180  89   6 504 151 352 113 210 105  41 310 124 462   1 363 206
## [137]  33 234 357 496 142 461 303 167 225 403 289 264  51 241 195  58  54
## [154] 140 125 160 431 245 454 242 118 443 197 386 404  10 317 162  34  52
## [171] 173 500 362 505 349 397  14 421 315 479 259 418 440  22 132 325 399
## [188] 174   2 116 382 450 226 102 215 424 239 204 319 270 318 411 491 101
## [205] 261 321 378 273 222 277 337 302 216 416 335  26 396 244 484 457 480
## [222]  28 361  88 276 260  15 223  40  27 231 111 214 340 467 182  60 217
## [239] 350 279 203  56  92 166  43 486 131   5 228 485 168 220  19  30 127
## [256] 267  80  99 345 148 331 181 149 430 254 169  23 492 229 379  70 183
## [273] 138 341 348 300 172 198 380 194 390 246 268 346 356 330  83 490 100
## [290] 444  12 468 437 284 301  29 202 410 487 339 477  49 251   4 394  42
## [307] 292 460 428 391   9 175 287 395  94  85 435  31 370 163  97 176 117
## [324] 465 355  98 427 368  63 456 451 158  91 324 332  48 464 283 213 219
## [341] 401 498 351 308 207 252 402 374 269 298 334 212 255  59 159 106 377
## [358] 253 412 360  38 388 286 248  50  72 297 164 178 488 417 199 436  71
## [375] 420 320 439  78 184 147 442 343 156  64 347 359 342 353 295 387 104
## [392] 366 192 466 328 333 126  77  45 497  81 221 364 227

Fitting the Linear Discriminant Analysis

First the linear discriminant analysis (LDA) is fitted to the train set. The new categorical variable crime is the target variable and all the other variables of the dataset are predictor variables.

After fitting we draw the LDA biplot with arrows.

## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2400990 0.2623762 0.2376238 0.2599010 
## 
## Group means:
##                   zn      indus         chas        nox          rm
## low       0.94892287 -0.9230350 -0.109974419 -0.8830429  0.43687127
## med_low  -0.05219905 -0.2991465 -0.012331882 -0.5945417 -0.12440342
## med_high -0.38808681  0.1900097  0.137785540  0.4030179  0.09310575
## high     -0.48724019  1.0170492 -0.009855719  1.0798759 -0.46200885
##                 age        dis        rad        tax    ptratio
## low      -0.9124045  0.8916117 -0.7095543 -0.7431447 -0.4846995
## med_low  -0.3729296  0.3895785 -0.5539047 -0.5069226 -0.0914510
## med_high  0.4264565 -0.3731486 -0.3920855 -0.2899811 -0.2416883
## high      0.8311714 -0.8755308  1.6388211  1.5145512  0.7815834
##                black       lstat        medv
## low       0.37806001 -0.77452047  0.51699127
## med_low   0.34514358 -0.13410974  0.02361544
## med_high  0.05428073  0.04502413  0.18603070
## high     -0.78893431  0.91433433 -0.67137558
## 
## Coefficients of linear discriminants:
##                 LD1         LD2         LD3
## zn       0.12353308  0.64682644 -0.92214263
## indus   -0.05284177 -0.21086983  0.51816379
## chas    -0.06560184 -0.01418336  0.11214638
## nox      0.45794294 -0.85770339 -1.36068152
## rm      -0.09372081 -0.11219485 -0.18602505
## age      0.30116870 -0.34629856 -0.08136403
## dis     -0.05803413 -0.32768984  0.20821236
## rad      2.96790538  1.01608937  0.15592961
## tax     -0.01265549  0.01995781  0.26969570
## ptratio  0.15686311 -0.09318475 -0.33204408
## black   -0.14185620  0.03787123  0.18751896
## lstat    0.28121206 -0.30226494  0.31267417
## medv     0.24796989 -0.48816558 -0.19567366
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9482 0.0386 0.0132
##   [1] 4 2 1 3 4 2 3 3 4 2 1 4 3 3 2 2 4 2 4 4 2 1 4 2 1 1 1 1 4 4 2 1 2 2 4
##  [36] 2 1 2 3 4 2 4 1 2 3 1 4 2 3 4 3 4 2 1 4 3 4 3 3 4 3 1 2 3 4 4 1 4 1 2
##  [71] 3 1 2 3 1 3 3 2 3 1 4 4 4 1 4 3 4 2 2 4 4 4 2 3 2 4 2 2 3 1 4 4 4 3 3
## [106] 2 1 2 1 2 3 4 2 4 2 1 3 3 2 4 1 1 1 1 3 1 2 3 2 1 3 2 4 1 4 2 3 3 4 2
## [141] 3 4 2 3 3 4 1 3 2 2 1 1 1 3 2 3 4 2 4 2 2 4 1 4 4 2 3 3 3 1 2 2 4 2 1
## [176] 4 3 4 3 4 3 4 4 3 3 3 4 2 1 2 4 4 3 2 3 4 2 1 3 2 2 4 2 2 3 2 4 2 3 2
## [211] 1 1 2 4 1 3 4 2 3 4 4 3 4 1 2 3 3 3 1 3 3 2 2 1 4 1 2 1 1 1 1 1 1 3 2
## [246] 3 3 1 3 3 3 2 3 3 3 3 2 1 1 3 1 1 3 4 3 3 3 2 3 4 2 2 3 1 1 1 3 1 4 1
## [281] 4 2 3 1 2 1 1 2 1 4 2 4 4 1 1 3 1 4 4 1 4 2 2 1 4 2 1 4 4 4 2 2 1 4 1
## [316] 1 4 3 4 3 2 1 2 4 1 2 4 4 2 4 4 3 1 3 1 2 4 1 2 2 4 3 1 1 2 2 4 4 3 2
## [351] 1 3 1 2 3 2 4 2 4 4 1 4 1 2 2 2 1 3 1 4 4 1 4 2 4 3 4 2 2 3 4 1 3 2 1
## [386] 4 1 1 1 4 2 4 1 3 2 1 2 2 2 3 1 3 4 3

Predicting the classes

##           predicted
## correct    low med_low med_high high
##   low       19      10        1    0
##   med_low    0      13        7    0
##   med_high   0       9       21    0
##   high       0       0        0   22

Prediction were quite good. There was some errors in the middle of the range but classes low and especially high were good. Only one correct representative of high class was predicted to med_low class.

K-means algorithm

I’m going to calculate what is the optimal number of clusters for Boston data. First I reload and scale the data. Variables need to be scaled to get comparable distances between observation.

##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv        
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865

Next I calculate the distances between observations and determinen the number of clusters.

One way to determine the number of clusters is to look how the total of within cluster sum of squares (WCSS) behaves when the number of clusters changes. WCSS was calculated from 1 to 15 clusters. The optimal number of clusters is when the total WCSS drops radically. It seems that in this case optimal number of clusters is two. However we are here to learn so I decided to analyse model with four clusters.

After determining the number of clusters I run the K-means alcorithm again.

It seems that when the data is divided to four clusters there is some clear differences in distriputions of several variables. Crim, zn, indus and blacks are variables were one can distinguish clear patterns between clusters. Some variables (rad & tax) show that sometimes 1 or 2 clusters make a clear distripution but observation of other two clusters are ambigious and there is no clear pattern to be regognised.

BONUS: LDA using clusters as target classes

After loading the Boston dataset I scale it to get comparable distances.

##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv             clust      
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063   Min.   :1.000  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989   1st Qu.:2.000  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449   Median :3.000  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   :2.674  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683   3rd Qu.:3.000  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865   Max.   :4.000

Original Boston dataset is now scaled and the result of K-means clustering is saved to the variable clust

LDA with the clusters

Next the LDA is performed and the biplot with arrows is created

## Call:
## lda(clust ~ ., data = scaled_Boston)
## 
## Prior probabilities of groups:
##         1         2         3         4 
## 0.2114625 0.1304348 0.4308300 0.2272727 
## 
## Group means:
##         crim         zn      indus       chas        nox         rm
## 1 -0.3912182  1.2671159 -0.8754697  0.5739635 -0.7359091  0.9938426
## 2  1.4330759 -0.4872402  1.0689719  0.4435073  1.3439101 -0.7461469
## 3 -0.3894453 -0.2173896 -0.5212959 -0.2723291 -0.5203495 -0.1157814
## 4  0.2797949 -0.4872402  1.1892663 -0.2723291  0.8998296 -0.2770011
##          age        dis        rad        tax     ptratio       black
## 1 -0.6949417  0.7751031 -0.5965444 -0.6369476 -0.96586616  0.34190729
## 2  0.8575386 -0.9620552  1.2941816  1.2970210  0.42015742 -1.65562038
## 3 -0.3256000  0.3182404 -0.5741127 -0.6240070  0.02986213  0.34248644
## 4  0.7716696 -0.7723199  0.9006160  1.0311612  0.60093343 -0.01717546
##        lstat        medv
## 1 -0.8200275  1.11919598
## 2  1.1930953 -0.81904111
## 3 -0.2813666 -0.01314324
## 4  0.6116223 -0.54636549
## 
## Coefficients of linear discriminants:
##                 LD1        LD2         LD3
## crim     0.18113078 -0.5012256 -0.60535205
## zn       0.43297497 -1.0486194  0.67406151
## indus    1.37753200  0.3016928  1.07034034
## chas    -0.04307937 -0.7598229 -0.22448239
## nox      1.04674638 -0.3861005 -0.33268952
## rm      -0.14912869 -0.1510367  0.67942589
## age     -0.09897424  0.0523110  0.26285587
## dis      0.13139210 -0.1593367 -0.03487882
## rad      0.65824136  0.5189795  0.48145070
## tax      0.28903561 -0.5773959  0.10350513
## ptratio  0.22236843  0.1668597 -0.09181715
## black   -0.42730704  0.5843973  0.89869354
## lstat    0.24320629 -0.6197780 -0.01119242
## medv     0.21961575 -0.9485829 -0.17065360
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.7596 0.1768 0.0636

Biplot shows that variables indus, zn and medv are the most influencial separators for the clusters.

Super-bonus

3D plot where observations color is the crime classes of the train set

3D plot where observations color is based on the K-means clusters.

Colors of the both plots is based to four classes. It seems that K-means plot shows the different clusters more clearly than the plot that is based on the crime classification.


RStudio exercise 5: Dimensionality reduction techniques

Introduction to the data